I am looking for a reference which explains how theta functions, algebraically independent meromorphic functions, and line bundles all fit together in the context of complex tori. More explicitly, given the right kind of Hermitian form on $\mathbb{C}^g$ with respect to some lattice $L$ I'd like an explicit construction of the g independent meromorphic functions on $\mathbb{C}^g/L$, which is something that Mumford, for example, does not give.

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Perhaps not in his "Abelian varieties" book, but certainly in his "Tata lectures on Theta" Mumford describes this setup. See Chapter II of book 1. Another reference is Birkenhake and Lange's book "Complex abelian varieties" (see Chapters 3 and 8, for instance). By the way, I don't think you generally construct just $g$ independent meromorphic functions; you construct a whole bunch more of them - enough to embed the abelian variety into projective space.